Question: Simplify; express your answer in exponential form. Assume $p\neq 0, q\neq 0$. $\dfrac{{(p^{5}q^{-5})^{-3}}}{{(p^{-2}q^{5})^{5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{5}q^{-5})^{-3} = (p^{5})^{-3}(q^{-5})^{-3}}$ On the left, we have ${p^{5}}$ to the exponent ${-3}$ . Now ${5 \times -3 = -15}$ , so ${(p^{5})^{-3} = p^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{5}q^{-5})^{-3}}}{{(p^{-2}q^{5})^{5}}} = \dfrac{{p^{-15}q^{15}}}{{p^{-10}q^{25}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-15}q^{15}}}{{p^{-10}q^{25}}} = \dfrac{{p^{-15}}}{{p^{-10}}} \cdot \dfrac{{q^{15}}}{{q^{25}}} = p^{{-15} - {(-10)}} \cdot q^{{15} - {25}} = p^{-5}q^{-10}$